UC-NRLF 


B    3    112 


ENGIN. 
LIBRARY 


NOTES 


ON    THE 


Graphics  of  Machine  Forces 


BY 


ROBERT    C.    H.   HECK,    M.E. 

PROFESSOR    OF    MECHANICAL    ENGINEERING 


WITH  39  ILLUSTRATIONS 


NEW    YORK 

I)   VAN  NOSTRAND  COMPANY 

23    MTRRAY   AND    27    WARREN    STREETS 

1910 


GIFT  OF 
Joseph  Le  Conte 


FN'GINEERING  LIBRARY 


The  D.  Van  Noftrand  Company 

intend  this  book  to  be  sold  to  the  Public 
at  the  advertised  price,  and  supply  it  to 
the  Trade  on  terms  which  will  not  allow 
of  reduction. 


NOTES 


ON    THE 


Graphics  of  Machine  Forces 


BY 


ROBERT    C.   H.   HECK,    M.E. 

i   OF    MECHANICAL   ENG 
IN    RUTGERS    COLLEGE 


\\ 
PROFESSOR   OF    MECHANICAL    ENGINEERING 


WITH  39  ILLUSTRATIONS 


NEW   YORK 

D.  VAN  NOSTRAND  COMPANY 
23   MURRAY  AND    27   WARREN   STREETS 

IQIO 


7V/7J- 


Copyright,  1910, 

BY 
D.  VAN   NOSTRAND    COMPANY 


>^o>~ 


THE  SCIENTIFIC   PRESS 

ROBERT    DRUMMOND    AND    COMPANY 

BROOKLYN,    N.   Y. 


PREFACE 


THESE  notes  are  intended  to  serve  as  text  for  a  graphical 
course  which  has  hitherto  been  based  upon  Herrmann's  "  Graph- 
ical Statics  of  Mechanisms."  To  the  latter  text  there  are  the 
objections  that  it  does  not  set  forth  the  fundamental  mechanical 
principles  clearly  enough  for  students  with  the  usual  degree  of 
effective  preparation,  and  that  it  wastes  entirely  too  much  space 
on  detailed  and  repeated  explanations  of  examples.  The  prob- 
lems in  this  course  are  in  the  shape  of  good-sized  drawings  of 
machines — on  sheets  about  20  in.  by  27  in. — which  are  repro- 
duced from  tracings  as  positive  prints,  with  dark  lines  on  a  light 
ground.  The  student  is  thus  saved  the  labor  of  mere  drawing, 
and  at  once  takes  up  the  force  determination.  On  the  drawing 
there  is  room  for  such  special  notes  and  suggestions  as  may  be 
called  for;  but  the  emphasized  purpose  is  to  have  the  student 
think  for  himself,  with  needed  help  and  suggestion  from  the 
instructor,  and  not  follow  a  ready  worked  example. 

In  section  J  are  added  some  special  force  constructions  which 
are  useful  chiefly  in  the  problems  of  graphical  dynamics,  but 
which  are  needed  to  round  out  the  presentation  of  graphical 
methods  for  determining  impressed  forces  in  machines. 

It  is  thought  that  the  title  here  used  is  more  appropriate  and 
descriptive  than  "  graphical  statics  of  mechanisms." 

R.  C.  H.  HECK. 

NEW  BRUNSWICK,  N.  J., 
May,  1910. 

iii 


M  2948 


CONTENTS 


5CTION  PARAGRAPHS  PAGE 

A.  GENERAL  CONDITIONS  OF  PROBLEMS  ...  i  to    2  i 

B.  FORCE  DIAGRAM  CONSTRUCTIONS        .       .       .       .  3  to  n  2 

C.  THE  ACTION  OF  FRICTION .  12  to  18  9 

D.  JOURNAL  FRICTION 19  to  26  13 

E.  THE  EFFICIENCY  OF  MACHINES          .       .       .       .  27  to  32  19 

F.  RESISTANCE  TO  ROLLING 33  to  35  23 

G.  TOOTHED-GEAR,  CHAIN,  AND  ROPE  RESISTANCES    .  36  to  42  25 
H.  BELT  TRANSMISSION 43  to  48  29 

I.  GENERAL  PROCEDURE 49  33 

J.  SPECIAL  FORCE  CONSTRUCTIONS          .       .       .       .  50  to  60  34 

K.  FRICTION  AND  LUBRICATION 6 1  to  64  40 


GRAPHICS    OF    MACHINE   FORCES 


A.  General  Conditions  of  Problems 

1.  In  order  to  determine,  by  the  simpler  im?ej,hp(Js\of  gra^hi^al 
analysis,  the  principal  forces  acting  in  inacriines,' we  make  the 
following  assumptions :  *  \  }  \  >;  \\\  ] 

(a)  The  weight  of  the  machine  members,  or  the  force  of 
gravity  acting  upon  them,  may  be  disregarded. 

(b)  The  force  of  inertia  need  not  be  taken  into  account:   this 
idea  involves  either  a  slow  running  of  the  machine  or  a  practi- 
cally uniform  motion  of  the  parts. 

A  great  many  problems  may  be  solved  with  quite  sufficient 
correctness  under  these  assumptions.  If  the  need  of  greater 
accuracy  or  their  increase  in  relative  magnitude  requires  that 
these  forces  (which  are  functions  of  mass  or  of  mass  and  motion) 
be  included,  the  solution  becomes  much  more  complicated. 
(See  Section  J.) 

2.  For  present  purposes,  the  machine  is  considered  as  made 
up  of  rigid  bodies,  or  of  members  which  are  fixed  and  definite 
in  form  so  far  as  the  forces  entering  into  the  problem  are  con- 
cerned.    These  are  all  impressed  forces,  imposed  through  the 
contact  of  one  machine  part  with  another.     The  forces  which  act 
upon  one  piece  get  at  each  other  and  come  into  equilibrium  through 
the  medium  of  internal  forces  or  stresses  within  the  body.     We 
pass  over  all  questions  as  to  the  magnitude  or  the  manner  of 
action  of  these  stresses,  and  use  simply  the  resultant  relations 
among  the  imposed  forces;    which  relations  are  the  same  as  if 
the  forces  acted  upon  and  met  in  a  single  material  particle. 


GRAPHICS  OF  MACHINE  FORCES 


B.  Force  Diagram  Constructions 

3.  In  every  case,  the  forces  on  any  member  of  the  machine 
are  to  be  considered  as  in  equilibrium.  The  requirements  for 
this  condition  are  as  follows: 

If  there  are  but  two  forces,  they  must  be  equal  and  opposite 
on  a  common  line  of  action. 

If  there  are  three  forces,  their  action-lines  must  meet  in  a 
common  point,  and  the  forces  must  form  a  closed  triangle. 

If  there  are  four  or  more  forces,  they  need  not  all  pass  through 
one  point,  but  they.irnist  be  reducible  to  the  three-force  case. 

•;With  piraBelrjforces,  we  must  apply  separately  the  require- 
ments that  the  algebraic  sum  of  the  forces  and  the  similar  sum 
,'of  ^'ieiVfmo'rri^tsJa.bout.any  center  shall  each  be  equal  to  zero. 


23 


1 0*3 


FIG.  i. — Three  Forces  on  Bell-crank. 

4.  In  the  three-force  example  shown  in  Fig.  i,  the  forces  on 
the  bell-crank  3  are  to  be  determined.  Rod-pull  (2  on  3)  is 
completely  known,  in  line  of  action,  direction,  and  intensity; 
rod-pull  (4  on  3)  is  known  in  line  of  action,  and  gives  the  inter- 
section P.  This  point  and  the  center  of  the  bearing  13  determine 
the  line  of  the  third  force,  the  bearing-pressure  (i  on  3).  Having 
used  the  drawing  of  the  machine  to  find  the  force  directions 
(that  is,  the  angular  positions  or  inclinations  of  the  lines  of  action), 
it  is  often  well  to  draw  the  force  diagram  separately,  as  in  Fig. 
i  b.  Here  the  line  AB  is  parallel  and  equal  to  (2  on  3),  BC  is 
parallel  to  (4  on  3),  and  AC  is  parallel  to  (i  on  3).  The  inter- 


FORCE  DIAGRAM  CONSTRUCTIONS  3 

section  at  C  fixes  the  length  or  value  of  the  latter  two  forces. 
Remember  that  for  equilibrium  the  direction-arrows  point  in 
circular  order  around  the  triangle. 

5.  The  two  typical  cases  of  determinate  conditions  under  the 
action  of  four  forces  will  now  be  taken  up — and  these  will  be 
considered  as  sufficiently  representing  all  cases  of  the  equilibrium 
of  more  than  three  forces  on  one  piece.  In  Fig.  2,  the  forces  on 
the  elevator  body  2  are,  the  load  Q,  the  upward  pull  P,  and  the 


t  ON 2 


G, 


FIG.  2. — Wall  or  Bracket  Elevator. 

two  guide-bar  pressures  G\  and  G2;*  all  the  action-lines  and'the 
intensity  of  one  force  Q  being  known.  Combine  the  forces  in 
pairs,  Q  with  G\,  P  with  G2.  For  equilibrium,  the  resultants 
RI  and  R2  must  balance,  hence  must  have  a  common  line  of 
action;  and  the  intersections  Oi  and  O2  determine  this  line. 
The  diagram  at  b  shows  how  RI  is  found  from  Q  and  GI,  then 
reversed  for  R2  and  resolved  into  P  and  G2. 


*  Here   friction   is  disregarded,   consequently   these  forces  are   perpendicular 
to  the  surfaces  in  contact. 


4  GRAPHICS  OF  MACHINE  FORCES 

6.  In  Fig.  3  the  piece  4,  which  is  made  up  of  the  wheels  and 
axle  of  a  locomotive,  is  subjected  to  four  forces  (besides  its 
weight  load,  not  here  included);  these  are,  the  pressures  of  the 
two  connecting-rods  2  and  3  upon  their  crank-pins,  the  pressure 
of  the  bearings  (parts  of  frame  i)  upon  the  axle,  and  the  tan- 
gential resistance  of  the  track  T.  Three  lines  of  action  are  known, 
but  for  the  location  of  the  fourth  force  there  is  only  one  determin- 
ing point.  The  problem  is  soluble  when  two  of  the  forces,  as 
(2  on  4)  and  (3  on  4),  are  completely  known;  whereupon  their 
resultant  R  can  be  found,  and  we  drop  to  the  three-force  deter- 
mination, with  the  intersection  P  as  the  second  point  required  to 


•4o«T  Ton4 

FIG.  3. — Driving  Action  in  Locomotive. 

fix  the  unknown  force-line.  Note  that  in  this  figure  the  force 
triangles  are  made  right  on  the  drawing  of  the  machine;  and  the 
force  R,  a  result  in  the  first  triangle,  is  moved  to  another  place 
on  its  line  of  action  in  order  to  enter  as  a  primary  quantity  into 
the  second  triangle. 

7.  The  equilibrium  of  three  parallel  forces  is  illustrated  in 
Fig.  4.  Always,  the  two  outer  forces  P  and  Q  act  in  the  same 
direction,  while  the  equilibrant  R,  equal  to  their  sum,  lies  some- 
where between  them.  The  only  moment  equations  that  need  be 
considered  are: 

With  origin  on  R,  Pa  =  Qb;         .......     (i) 

With  origin  on  P,  Ra=Qc;         (2) 

With  origin  on  Q,  Rb=Pc.          .     .     .     .     .     .     .     (3) 


FORCE  DIAGRAM  CONSTRUCTIONS  5 

All  of  these  are  expressed  graphically  in  the  triangles  obtained 
by  drawing  the  parallel  cross-lines  AC,  DF,  and  GK,  and  the 
diagonal  GF.  Considering  the  first  relation,  for  instance,  we 
have  from  the  similar  triangles  ABG,  CBF: 


or 


CF:CB::AG:AB, 

P:b::Q:a, 


whence 


FIG.  4. — Parallel  Force  Relations. 

For  equal  moments,  the  forces  must  be  inversely  as  their 
own  moment-arms,  or  directly  as  the  opposite  arms.  In  effect, 
in  Fig.  4,  each  force  is  transferred  to  the  line  of  the  other  force, 
P  to  FC,  Q  to  AG,  and  there  is  then  a  direct  proportion  to  the 
arms  BC  and  AB.  This  idea  of  interchanging  the  forces  for 
purposes  of  graphical  construction  is  used  in  every  case;  thus 
for  the  second  relation,  with  origin  of  moments  on  line  of  P, 
we  put  R  at  KF  and  Q  at  BH  and  have  the  forces  in  the  ratio 
of  the  distances  GK  and  GH.  The  cross-lines  AB,  DF,  and  GK 
need  not  be  perpendicular  to  the  force-lines;  their  parallelism 
is  the  essential  thing. 

8.  The  typical  problems  in  parallel  forces  are  represented 
in  Fig.  5.  In  each  diagram,  full  lines  show  originally  known 


6 


GRAPHICS  OF  MACHINE  FORCES 


quantities,  dotted  lines  the  results  got  by  construction.  The 
figures  are  lettered  in  such  a  manner  that  the  alphabetical  order 
indicates  the  order  in  drawing  the  lines.  The  four  cases  may  be 
briefly  set  forth  as  follows: 

Case  I. — The  three  lines  of  action  and  an  outer  force  P  are 
known:  draw  AB  and  CD,  to  carry  P  over  to  line  of  ();  diagonal 
DEF  determines  Q  in  AF  and  R  in  CF,  and  line  FG  transfers 
these  forces  to  their  proper  lines. 

Case  II.- — The  three  lines  of  action  and  the  inner  force  R  are 
known:  draw  AB-AC  and  DE-DF,  to  carry  R  to  line  of^Q; 


R 


III. 


FIG.  5. — Problems  in  Parallel  Forces. 


diagonal  CGE  divides  R  into  AG  or  Q  and  GD  or  P,  and  line 
GH-GK  transfers  the  forces  to  their  own  lines. 

Case  III.- — Two  forces  in  same  direction  are  completely 
known,  and  third  (middle)  force  (known  in  intensity,  as  the  sum 
of  the  other  two)  is  to  be  located:  draw  AB,  CD,  and  EF,  to 
carry  forces  to  opposite  lines,  P  to  DB,  Q  to  AF;  then  diagonal 
FGD  locates  at  G  an  origin  about  which  P  and  Q  have  equal 
moments,  which  is  therefore  a  point  on  line  of  R. 

Case  IV. — Two  forces  in  opposite  directions  are  completely 
known,  and  the  third  (the  outer  force  beyond  the  larger  of  the 


FORCE  DIAGRAM  CONSTRUCTIONS  7 

two,  and  known  in  intensity  as  their  difference)  is  to  be  located: 
draw  AB,  CD,  EF,  and  diagonal  FDG  to  meet  AB  produced 
at  G;  the  latter  point  locates  the  line  of  Q,  and  DH  and  EK 
carry  the  force  over  to  this  line. 


FIG.  6. — Construction  for  Distant  Intersection. 

9.  When  two  forces  are  nearly  parallel,  so  that  their  inter- 
section falls  far  off  the  drawing  board,  it  is  necessary  to  find 
the  direction  of  the  third  force  without  actually  drawing  it  to 
the  intersection.  In  Fig.  6,  AB  and  CD  are  the  known  force- 
lines,  and  the  third  force  is  to  pass  through  the  point  E.  Draw 
a  cross-line  AC  through  E  and,  at  a  convenient  distance,  a 
parallel  cross-line  BD.  The  third  force-line  EF  must  divide 
BD  in  the  same  ratio  as  AC.  From  A  draw  AD'  (at  any  con- 
venient angle)  equal  to  BD,  join  CD',  and  draw  EF'  parallel 
to  CD';  then  lay  off  point  F'  on  BD,  at  F,  and  draw  EF. 


FIG.  7. — Geometrical  Constructions. 

10.  Another  scheme  for  dividing  the  second  cross-line  in  the 
same  ratio  as  the  first,  which  involves  no  measurement  or  transfer 
of  lengths,  is  shown  in  Fig.  7.  For  the  first  case,  with  the  point  E 
between  the  known  lines,  draw  the  diagonal  CB;  then  EG,  parallel 
to  AB,  carries  the  ratio  (AE:EC)  to  (BG:GC);  and  drawing 


8 


GRAPHICS  OF  MACHINE  FORCES 


GF  parallel  to  CD  makes  (BF:FD)  equal  (BG:GC).  To  apply 
this  exact  scheme  when  E  lies  outside  the  known  lines,  we  should 
have  to  draw  EG'  parallel  to  AB,  meeting  CB  produced,  then 
draw  G'F  parallel  to  CD.  It  is  more  convenient,  and  tends  to 
better  accuracy,  again  to  make  the  construction  on  the  middle 
line,  now  the  known  line  CD:  draw  CG  parallel  to  AB,  join 
GD,  and  draw  EF  parallel  to  GD.  The  essential  thing  is  to 
have  a  line  parallel  to  the  base  of  a  triangle,  so  as  to  divide  the 
sides  in  the  same  ratio. 

ii.  When  three  intersecting  forces  are  not  far  from  parallelism, 
the  force  triangle  becomes  very  flat,  as  at  diagram  b  in  Fig.  8, 
and  the  intersection  U  of  the  sides  representing  the  unknown 


FIG.  8.— The  Flat  Force  Triangle. 

forces  is  rather  indefinite.  For  accurate  determination  of  these 
forces,  the  moment  relation  must  supplement  the  force  triangle. 
One  method  is  to  take  an  origin  on  the  line  of  one  unknown 
force,  as  at  G,  draw  and  measure  the  perpendicular  moment- 
arms  GH  and  GJ,  measure  the  force  P,  and  multiply  and  divide 
in  the  relation, 

Rb=Pc,          (4) 

Or,  as  indicated  at  c,  a  regular  parallel-force  construction  may 
be  made  for  one  force,  as  R.  In  either  case,  after  a  second  force 
has  been  found  we  go  back  to  the  force  triangle  and  use  it  to 
get  the  third  force. 


THE   ACTION    OF  FRICTION 


C.  The  Action  of  Friction 

12.  In  Fig.  9  is  represented  a  block  2  which  is  pressed  down 
upon  the  plane  i  by  a  force  N,  normal  to  the  surfaces  in  con- 
tact, and  is  supported  by  the  equal  and  opposite  upward  re- 
action of  the  plane  i.  If  the  pressure  between  the  surfaces  is 
uniformly  distributed  over  the  contact-area,  as  indicated  by  the 
diagram  at  the  right,  the  equivalent  concentrated  force  N  will 
act  at  the  geometrical  center  of  this  area.  If  the  block  is  made 
to  move,  as  in  the  direction  of  the  motion-arrow,  there  will  be 
developed  between  the  surfaces  a  frictional  resistance  F,  opposing 
the  movement.  This  frictional  force  F  bears  to  the  normal  pres- 


FIG.  9. — Resistance  to  Sliding. 

sure  N  a  ratio  which  is  known  as  the  coefficient  of  friction.  As 
in  every  case,  there  is  here  the  inevitable  condition  of  an  equal 
and  opposite  action  and  reaction:  the  friction  force  F1  on  2 
opposes  the  movement  of  2  on  i,  while  the  reaction  F2  on  1  sim- 
ilarly opposes  the  movement  of  i  relative  to  2. 

13.  In  Fig.  9  only  the  conditions  right  at  the  contact  sur- 
faces are  considered,  and  nothing  is  shown  as  to  the  detail  of 
the  external  forces  on  the  block  2.  The  complete  representation 
of  a  simple  case  is  given  in  Fig.  10.  Block  2  is  pulled  downward 
by  its  own  gravity  force  W,  which  necessarily  passes  through  the 
center  of  gravity  G;  and  the  force  P,  which  moves  the  block, 
acts  along  the  line  CD,  parallel  to  the  plane.  To  bring  the 
block  just  to  the  eve  of  motion,  or  to  maintain  a  uniform  motion, 
P  must  equal  F\  and  the  resultant  of  W  and  P  is  the  pressure 


10 


GRAPHICS  OF  MACHINE  FORCES 


of  2  on  i,  balanced  by  that  of  i  on  2.  The  latter  force  has  F^  on  2 
as  one  component,  while  the  other  is  N,  equal  and  opposite  to 
W.  Since  P  and  F  are  not  along  the  same  line,  they  form  a 
turning  couple;  but  their  moment  Pa  is  balanced  by  the  equal 
and  opposite  moment  Wb  of  the  couple  W  and  N.  Further, 
since  the  total  pressure  (2  on  i)  or  (i  on  2)  is  not  central  at  the 
contact  surface,  the  actual  distributed  force  will  be  non-uniform, 
as  shown  by  the  pressure  diagram  at  the  right. 


FIG.  10. — Block  Sliding  on  Plane. 

14.  The  arrangement  outlined  in  Fig.  n  is  frequently  used 
to  illustrate  the  action  of  sliding  friction,  and  serves  especially 
to  give  fuller  meaning  to  the  expression  "  angle  of  friction." 
The  gravity  force  W  is  resolved  into  components,  N  normal  and  P 
parallel  to  the  plane;  and  the  inclination  of  the  plane  is  just 
enough  to  make  the  active  component  P  equal  to  the  frictional 
resistance  F — by  "  active  "  component  is  meant  the  one  in  the 
direction  in  which  the  body  is  capable  of  moving  under  its  con- 
ditions of  constraint  of  motion.  The  angle  a,  which  the  plane 
AB  makes  with  the  horizontal  base  BC,  or  which  the  total  pres- 
sure W  makes  with  the  normal  force  A7,  is  called  the  angle  of 
friction;  and  the  coefficient  of  friction  is  the  tangent  of  this 
angle,  or 

*     F 

/,*==_  =  tana .     ..,;,.    ,     (5) 

*  Greek  mu,  same  as  m. 


THE  ACTION  OF  FRICTION 


11 


15.  Suppose  that  under  the  conditions  of  Fig.  9  or  Fig.  10  a 
force  P  is  applied  which  is  less  than  the  total  friction  /*N;  then 
only  so  much  of  the  frictional  resistance  will  come  into  play  as 
is  needed  to  balance  P.  As  P  increases,  the  frictional  force  be- 


FIG.  ii. — Sliding  on  Inclined 
Plane. 


2  ON  I 


FIG.  12. — Limits  of  Frictional 
Resistance. 


tween  the  surfaces  also  increases,  up  to  the  limit  //N;  when 
this  limit  is  reached,  motion  is  ready  to  begin.  If  P  becomes 
greater  than  F,  the  line  of  resultant  pressure  of  i  on  2  does  not 
swing  beyond  the  angle  a,  as  shown  at  AB  or  AC  in  Fig.  12, 
but  the  excess  of  P  over  F  is  a  free  or  unbalanced  force,  acting 
to  accelerate  the  moving  body.  In  a  reversal  of  motion,  as  by 
an  engine  crosshead  at  the  end  of  the  stroke, 
the  line  of  pressure  swings  through  the  angle 
20:,  say  from  AB  to  AC. 

T 6.  The  first  question  to  be  answered  in 
any  case  of  sliding  friction  in  a  machine  is, 


ZONt 


To  which  side  of  the  normal  is  the  pressure-  FIG   I3._Inclination  of 
line  inclined?    The  general  principle  of  all        the  pressure-line. 
friction  action  is  that  friction  opposes  rela 

tive  movement;  consequently,  the  pressure  acting  upon  any  piece 
at   the   contact  surface   must  have    a  component   against    the 


12 


GRAPHICS  OF  MACHINE  FORCES 


movement  of  that  piece.  The  simple  "  rule  of  thumb "  is 
illustrated  in  Fig.  13.  At  about  the  place  where  the  force-line 
will  cross  the  contact-surface,  Crosshatch  a  little  block  on  each 
piece  to  emphasize  the  contact,  and  draw  arrows  indicating  the 
respective  directions  of  relative  movement  of  the  two  pieces; 
then  draw  an  inclined  line  which  joins,  or  tends  to  join,  the 
tails  of  the  arrows.  This  line  shows  the  direction  in  which  the 
pressure-line  slants  away  from  the  normal  to  the  surfaces. 

17.  The  machine  outlined  in  Fig.  2  and  reproduced  in  Fig. 
14  serves  as  a  very  good  example  of  the  effect  of  sliding  friction. 


'  FIG.  14. — Wall  Elevator,  with  Friction. 

The  first  step  is  to  sketch  in  at  A  and  D  the  determination  of 
Fig.  13,  purposely  exaggerating  the  angle  of  inclination;  then 
the  action-lines  of  the  guiding  pressures  GI  and  G2  are  definitely 
drawn  by  measuring  off  along  the  normals  convenient  lengths 
AB  and  DE  and  erecting  perpendiculars,  BC  =  //XAB  and 
EF  =  /*XDE.  The  intersections  Oi  and  O2  now  fix  the  re- 
sultant line,  and  the  force  diagram  takes  the  form  shown  in  full 
lines  at  b.  With  this  is  drawn  in  dotted  lines  the  diagram  for 
downward  motion  of  the  elevator,  and  in  dot-and-dash  lines 
the  diagram  for  the  ideal  case  of  no  friction,  reproduced  from 
Fig.  2.  In  downward  movement  the  weight  Q  becomes  the 


JOURNAL  FRICTION 


13 


driving  force,  while  P,  now  the  hold-back  or  brake  force,  serves 
as  resistance. 

1 8.  It  is  of  interest  to  set  forth  a  reason  for  the  fact  that, 
in  Fig.  14,  the  guiding  forces  G\  and  G^  and  consequently  the 
friction  on  the  guide-bar,  are  greater  in  downward  than  in  up- 
ward motion- — that  is,  a  reason  with  a  more  fundamental  basis 
than  the  mere  appearance  of  this  result  in  the  force-diagram 
determination.  In  Fig.  15  are  given  diagrams  of  all  the  vertical 
forces  on  the  elevator  body,  the  third  force  F  being  the  resultant 
of  the  two  frictions  and  equal  to  the  difference  between  P  and  Q. 
The  resultant  turning  moment  of  the  three  forces  must  be  balanced 
by  the  couple  made  up  of  the  normal  components  of  G\  and  G%. 


FIG.  15. — The  Moment  of  Friction  in  Fig.  14. 

Taking  the  center  of  moments  on  the  line  of  force  P,  we  see  that 
in  case  a  the  moment  Fn  acts  against  Qm,  while  in  case  b  these 
two  moments  act  together.  This  example  shows  the  advantage 
of  using  other  lines  of  reasoning  to  supplement,  and  sometimes 
to  check,  the  direct  graphical  determination. 


D.  Journal  Friction 

19.  The  turning  of  a  journal  in  its  bearing  is  nothing  but  a 
sliding  of  curved  surfaces  upon  each  other,  but  the  conditions 
as  to  pressure-distribution  are  much  less  simple  than  with  flat 
surfaces.  In  Fig.  i6a  the  journal  is  supposed  to  fit  the  bearing 
with  a  snug  running  fit,  close  but  not  forced;  then  under  the 
action  of  the  force  AT"  the  pressure  would  be  distributed  about 
as  shown  by  the  diagram.  An  actual  bearing  is  somewhat  loose 


14 


GRAPHICS  OF  MACHINE  FORCES 


on  the  journal;  wear  in  service  tends  toward  a  uniform  distribu- 
tion of  pressure;  and  between  the  surfaces  there  is  an  elastic 
film  of  lubricant.  For  these  reasons  the  curve  of  distribution 
will  probably  take  the  form  at  b  in  Fig.  16;  it  is  more  nearly 
uniform  over  a  considerable  arc  at  the  bottom,  but  does  not  cover 
the  entire  half-circumference.  The  essential,  fact,  in  any  case, 
is  that  the  total  pressure  between  the  surfaces  is  greater  than 
the  resultant  N- — this  because  the  oblique  pressure  away  from 
the  resultant-line  has  only  a  component  for  or  against  N.  With 
the  same  load  and  the  same  coefficient  there  would  be,  therefore, 
more  frictional  resistance  in  a  bearing  than  under  a  slide-block. 


ION  2 


FIG.   16. — Pressure  between 
"Journal  and  Bearing. 


2owl 


FIG.  17. — Journal  Friction. 


20.  Actually,  the  coefficient  of  friction  is,  for  the  same  mate- 
rials and  lubricant,  quite  a  good  deal  less  in  the  bearing  than  on 
the  slide,  chiefly  because  the  lubrication  is  so  much  more  effective, 
as  explained  in  paragraph  63,  following.  Further,  the  coeffi- 
cient as  experimentally  determined  includes  the  influence  of 
obliquity  of  surfaces,  since  the  experiments  are  made  with  bearings 
of  actual  form.  Practically,  we  treat  the  problem  as  if  the  pres- 
sure and  its  resulting  friction  were  concentrated  upon  a  narrow 


JOURNAL  FRICTION  15 

flat  surface  which  is  central  on  the  main  line  of  thrust.  This 
condition  is  depicted  in  Fig.  17,  where  the  fundamental  rela- 
tions are  closely  analogous  to  those  in  Fig.  9.  It  is  assumed  that  all 
special  details  in  the  action  of  pressure  and  friction  have  exerted 
their  influence  in  helping  to  determine  the  friction  -coefficient 
0  (phi). 

21.  In  Fig.  17,  the  pressure  P  makes  with  the  normal  N 
the  friction  angle  «;  and  the  coefficient  is,  as  before,  (f>  =  F/N= 
tan  a.  To  oppose  rotation  about  the  journal-axis  O,  the  fric- 
tion al  force  F  exerts  the  resisting  moment, 


(6) 


In  order  that  the  force  P,  which  is  equal  to  N/cos  a,  shall  have 
this  moment,  it  must  satisfy  the  relation, 


(7) 


cos  a 
and  pass  the  center  of  the  journal  at  the  distance, 

r=(j>Rcosa  .......     (8) 

The  same  relation  is  easily  derived  geometrically;   in  the  triangle 
ABO, 


cos  o: 


cos  a  =  6R  cos  a.      .     .     .     (Q) 


Since  the  angle  a:  is  always  small,  we  disregard  the  factor  cos  a, 
and  use  for  the  friction  circle  the  radius, 


(10) 


With  the  line  of  total  pressure  tangent  to  the  friction  circle,  in- 
stead of  passing  through  the  center,  the  pressure  diagram  will 
be  distorted  from  the  symmetrical  form  in  Fig.  16:  it  will  be 
heavier  on  the  side  where  the  journal  surface  is  descending,  or 
where,  to  a  slight  degree,  the  journal  tends  to  climb  in  the 
bearing. 


16  GRAPHICS  OF  MACHINE  FORCES 

22.  Examples  illustrating  the  use  of  the  friction  circle  are 
given  in  Fig.  18,  with  the  simple  link-work  fully  outlined  at  a 
as  basis.     The  single  force-line  along  the  connecting  link  3  is 
to  be  located,  and  the  question  is,  To  which  side  of  the  friction 
circle  will  this  line  be  tangent  at  each  joint?     Without  friction, 
the  line  will  go  through  the  centers  of  the  two  journals.     In  case 
a,  the  rod  is  in  a  state  of  tension,  or  is  pulling  down  on  the  arm 
2,  hence  the  "  contact "  will  be  on  the  top  of  the  upper  pin — 
and  on  the  bottom  of  the  lower  pin.     Noting  that  the  angle  between 
links  2  and  3  is  becoming  less  or  "  closing,"  we  see  that  3  will 
have  left-hand  or  anti-clockwise  rotation  with  reference  to   2, 
or  that  2  will  turn  toward  the  right  on  3 ;  and  the  direction  arrows 
are  drawn  accordingly.     A  line  joining  the  tails  of  these  arrows, 
as  in  Fig.  13,  shows  to  which  side  of  the  friction  circle  the  final 
force-line  will  pass.     The  same  determination  is  made  at  the 
lower  joint,  where  the  relative  movement  is  the  reverse  of  that 
at  the  top;   and  the  force-line  is  drawn  tangent  to  the  two  fric- 
tion circles  on  the  sides  indicated. 

23.  For  the  preliminary  determination  of  the  side  to  which 
the  true  line  of  force  is  deflected  from  the  normal,  it  is  proper 
to  center  the  contact  on  the  force-line  for  the  case  of  no  friction 
(it  is  understood  that  the  force-action  without  friction  will  have 
been  worked  out  before  the  case  with  friction  is  taken  up). 
Remember,  however,  that  this  shows  only  the  direction  of  the 
deflection  a,  and  does  not  fix  the  radial  line  from  which  this 
angle  «  will  finally  be  measured.     The  friction  circle  expresses 
geometrically  the  condition  to  be  met  by  the  force-line  at  this 
one  journal-bearing;    if  the  line  is  tangent  to  the  circle,  it  will 
make  the  proper  angle  a  with  a  radius  drawn  to  the  point  where 
it  crosses  the  circumference.     To  fix  the  line,  another  determinant, 
outside  of  the  single  bearing,  is  required. 

24.  In  parts  b  and  c  of  Fig.  18  are  shown  two  out  of  a  number 
of  possible  variations  from  case  a.     At  b  there  is  an  interchange 
of  pin  and  eye  between  the  links  at  each  joint,  as  compared  with 
a;    but  the  final  result,  in  the  location  of  the  force-line,  is  un- 
changed.    At  c,  however,  link  4  is  made  driver  and  the  direction 


JOURNAL  FRICTION 


17 


of  motion  is  kept  the  same,  thus  changing  the  stress  in  rod  3 
from  tension  to  compression;  and  now  the  force-line  is  reversed 
in  tangency  at  both  joints.  The  following  general  rules  govern 
these  and  other  changes: 

To  interchange  journal  and  bearing  or  pin  and  eye  at  a  joint 
(or  to  be  uncertain,  in  a  problem,  as  to  which  part  belongs  to 
which  piece)  will  not  affect  the  force-line  at  this  joint,  provided 
that  the  contacts  and  motions  are  correctly  represented  for  the 
existing  arrangement. 

To  reverse  either  direction  of  force  or  direction  of  motion 
at  a  joint  will  swing  the  force-line  to  the  other  side  of  the  fric- 


2oN3 


40N3 


FIG.  18. — Use  of  the  Friction  Circle. 


tion  circle,  but  to  reverse  both  together  will  leave  it  unchanged. 
The  two  changes  act  like  minus  signs  in  an  algebraic  multipli- 
cation, one  effecting  a  reversal,  two  neutralizing  each  other. 

25.  A  very  useful  check  upon  the  detailed  determination  of 
the  side  of  tangency  to  the  friction  circle  (that  is,  upon  the 
method  of  Fig.  17)  is  got  by  applying  the  general  principle  that 
friction  must  hinder  the  motion.  Thus  in  Fig.  i8a,  "  driver  " 
2  is  made  to  turn  about  the  center  12  by  some  driving  force  not 
shown  on  the  figure,  and  the  force  (3  on  2)  acts  as  a  resistance  to 
motion;  this  resistance  is  given  a  greater  effect  by  moving  it 
outward  from  12,  or  increasing  its  moment-arm  from  that  center. 


18 


GRAPHICS  OF  MACHINE  FORCES 


At  the  lower  joint,  however,  force  (3  on  4)  is  a  driving  force,  and 
the  hindering  effect  of  friction  is  shown  by  the  decrease  in  its 
moment-arm  (from  14)  due  to  tangency  on  the  inner  side  of  the 
friction  circle.  This  check  is  not  always  so  obvious  or  so  easy 
to  apply  as  in  this  example,  but  it  should  constantly  be  kept 
in  mind. 

26.  For  the  purpose  of  finding  the  relative  motion  in  a  turning 
joint,  the  scheme  illustrated  in  Fig.  19  is  very  effective.  Con- 
sider joint  B :  the  lines  AB  and  CB  (of  constant  length)  form  two 
sides  of  a  triangle,  with  AC  as  base.  Whether  AC  will  increase 
or  decrease  as  the  mechanism  moves  in  the  direction  of  the  arrow. 


FIG.   19. — Determination  of  Direction  of  Motion. 

determines  whether  the  angle  ABC  is  opening  or  closing.  Point 
C  is  moving  in  a  direction  perpendicular  to  the  arm  CD;  and 
since  this  direction  makes  an  acute  angle  with  AC,  we  see  that 
the  latter  is  decreasing.  For  the  joint  C,  with  DB  as  the  variable 
base,  the  corresponding  angle  is  obtuse,  hence  DB  is  increasing 
and  the  angle  opening.  If  C  were,  for  a  particular  position  of  the 
mechanism,  moving  in  a  direction  perpendicular  to  AC,  it  would 
indicate  that  there  was,  for  the  instant,  no  turning  at  all  in  the 
joint  at  B. 


THE  EFFICIENCY  OF  MACHINES  19 


E.  The  Efficiency  of  Machines 

27.  The  efficiency  of  a  machine  is  the  ratio  of  the  useful 
work  delivered  to  the  total  work  put  in.  Let  P  stand  (as  hereto- 
fore) for  the  driving  force  and  Q  for  the  useful  resistance;  and 
while  P  moves  a  distance  p*  let  Q  move  a  distance  q*,  the  ratio 
of  p  to  q  being  determined  wholly  by  the  proportions  and  con- 
figuration of  the  mechanism.  Further,  let  the  total  friction  of 
the  machine  be  combined  in  a  single  force  F,  which  is  overcome 
through  the  distance/.  Then  the  general  relation  is, 

'.,-.     .....     (ii) 


or,  work  put  in  equals  useful  work  plus  work  wasted  against 
friction.     The  efficiency  is, 


28.  The  graphical  methods  with  which  we  are  now  concerned 
determine  forces,  but  not  distances  or  velocities.  It  is  therefore 
desirable  to  be  able  to  express  the  efficiency  e  as  a  ratio  of  forces 
rather  than  of  work  quantities.  This  end  is  attained  with  the 
help  of  the  ideal  case  without  friction,  for  which  we  have  the 
relation, 

Pp=Qq  .....   •   •   •    (13) 

Suppose  that  we  start  P  as  the  known  force,  and  work  through 
to  Q.  The  value  without  friction,  which  we  call  <2o,  will  be 
greater  than  the  actual  value  Q,  and  the  efficiency  will  be, 

_Qg_Qq_Q_  .     . 

---- 


*  These  are  effective  distances,  measured  in  the  directions  of  the  forces.  If 
the  force  is  oblique  to  the  path  of  the  point  of  application,  we  may  use  the  com- 
ponent of  motion  along  the  force-line  for  p  or  q,  or  we  may  use  the  total  motion 
and  with  it  the  force-component  along  the  path  as  P  or  Q. 


20  GRAPHICS  OF  MACHINE  FORCES 

If,  on  the  other  hand,  we  have  Q  known  and  work  back  to  P, 
the  actual  P  (with  friction)  will  be  greater  than  the  ideal  P0, 
and  the  efficiency  will  be, 

Qq    Pop    Po 
=  P-P=JJ  =  -P 

29.  Some    machines,    notably   hoisting    machines    (in .  which 
gravity  acts  as  the  load-force  Q),  can  run  backward  with  the 
same  set  of  forces  as  in  forward  running,  but  with  Q  now  acting 
as  driving  force  and  P  as  resistance.     To  distinguish  the  forces 
acting  under  this  condition,  we  bracket  them  thus,    (P),    (Q): 
the  work  equation  takes  the  form, 

(Q)q=(P)P+(F)f,      (16) 

and  the  expressions  corresponding  to  equations  (14)  and  (15)  are, 

«=!•  <->=£ <"> 

The  forward  efficiency  e  is  always  less  than  unity,  lying  some- 
where between  zero  and  one;  the  backward  efficiency  (e)  may 
pass  through  zero  and  become  negative,  and  on  this  negative 
side  may  become  greater  than  one.  This  peculiar  state  of  af- 
fairs can  best  be  understood  with  the  help  of  an  example. 

30.  The  mechanism  in  Fig.  20,  outlined  for  each  case  at  a, 
has  for  its  moving  parts  the  wedge  2  and  the  vertical  slide-block 
3,  with  the  load  Q.     In  forward  motion,  the  w^edge  is  pushed 
toward  the  right  and  the  load  lifted;    in  backward  motion,  the 
wedge  is  withdrawn  and  the  load  descends.     The  two  cases  are, 
I,  wedge  steep  or  blunt,  backward  efficiency  positive;   II,  wedge 
flat   or   sharp,    backward   efficiency   negative.     On   diagrams   a 
the  various  force-directions  are  indicated,  but  the  lines  are  drawn 
so  as  to  keep  clear  of  each  other,  and  no  attempt  is  made  to  get 
the  proper  intersections.     As  in  Fig.  14,  dot-and-dash  lines  are 
used  for  forces  without  friction,*  full  lines  for  forward -motion 

*  Without  friction,  the  forces  are  the   same  in  either  forward  or  backward 
motion. 


THE  EFFICIENCY  OF  MACHINES 


21 


forces,  dotted  lines  for  the  case  of  backward  motion.  Diagrams 
b,  with  all  the  forces  marked,  should  require  little  explanation: 
a  force  triangle  with  Q  as  the  known  side  is  first  drawn  for  piece 
3;  then  force  (2  on  3),  reversed  to  (3  on  2),  is  the  base  for  the 
construction  of  the  triangle  for  piece  2. 

31.  The  essential  difference  between  the  two  cases  in  Fig.  20 
lies  in  the  opposite  directions  of  the  force   (P).     In  case  I  it 


P- 


I. 


P  <--___ 


(P)  P0  P 

FIG.  20. — Reversal  of  Backward  Efficiency. 


points  in  the  same  direction  as  PO  or  P;  and  Q,  driving  the 
machine  backward,  does  a  "  useful  "  work  in  overcoming  the 
resistance  (P).  In  case  II  the  friction  effect  is  so  great,  relative 
to  the  forces  along  the  wedge,  that  Q  alone  cannot  produce  back- 
ward motion.  Force  (P),  instead  of  holding  the  wedge  in  place 
and  acting  as  a  brake-force,  must  turn  around  and  help  Q.  The 
full  effect  exerted  by  Q,  if  expressed  as  a  horizontal  force  on  the 


22  GRAPHICS  OF  MACHINE  FORCES 

wec'ge  2,  is  equal  to  PO.  If  the  friction  in  the  machine  is  just 
enough  to  balance  Q  or  to  neutralize  PO,  the  backward  efficiency 
(e)  is  zero.  If,  as  here,  the  friction  exceeds  this  amount,  or  (P) 
has  to  help  Q,  this  efficiency  becomes  negative.  In  a  word,  the 
work  Qq  is  supplied  at  what  is  now  the  input  end  of  the  machine; 
but  to  make  it  move  at  all  the  further  work  (P)p  must  be  sup- 
plied at  the  output  end.  In  the  sense  of  a  normal  output,  this 
work  (P)p  is  negative,  hence  the  minus  sign  for  (e). 

32.  A  machine  which  has  so  much  friction  that  it  will  not  run 
backward  under  the  action  of  its  load  is  said  to  be  self-locking. 
With  this  property  goes  a  low  efficiency  in  forward  running. 
Ccnsider  the  case  where  the  friction-work  (F)f  is  just  equal  to 
Pop  or  Qq,  and  assume  that  in  forward  running  the  lost  work 
Ff  is  the  same  as  (F)f.  Then  from  equation  (n)  we  have, 

Pp  =  Qq  +  Ff=2Qq,         ....         (18) 

and  the  value  of  e  is  one-half  or  50  per  cent.  If  (F)f  is  greater 
than  Qq,  e  will  fall  below  this  limit.  The  difference  between 
(F)f  and  Ff  will  be  small,  with  a  general  probability  that  Ff 
will  be  the  larger  quantity,  because  the  forces  in  the  machine  are 
likely  to  be  a  little  greater  in  forward  than  in  backward  running. 
The  self-locking  machine  has  the  advantages  of  a  high  velocity- 
ratio  of  P  to  Q- — for  it  is  because  PO  is  relatively  small  that  fric- 
tion can  overbalance  it — and  is  usually  a  simple  device  for  lifting 
a  big  load  with  a  small  driving  force;  also,  it  makes  a  safe  hoist, 
because  no  brake  is  needed  to  hold  up  the  load.  The  fact  that 
at  least  half,  and  probably  more,  of  the  work  put  in  will  be  ex- 
pended in  overcoming  friction  is,  however,  a  decided  drawback. 
The  most  commcn  examples  of  this  type  of  machine  are  the  screw 
and  the  worm  and  worm-wheel,  both  derivatives  from  the  wedge 
mechanism  in  Fig.  20. 


RESISTANCE  TO  ROLLING 


23 


F.  Resistance  to  Rolling 

33.  When  a  heavy  or  heavily  loaded  roller  rests  en  a  plane 
surface,  there  is  always  some  elastic  compression  of  both  roller 
and  plane  at  the  contact.  This  contact  does  not  really  exist 
along  a  line,  but  is  spread  over  a  narrow  surface;  the  character 
of  the  pressure-distribution  is  indicated  in  Fig.  21,  but  with 
the  width  of  contact  tremendously  exaggerated.  If  the  roller 
advances,  there  is  a  continual  compression  of  the  surfaces  on 
the  side  of  advance  and  a  release  or  expansion  on  the  opposite 
side;  the  net  result  is  a  resistance  to  rolling,  which  can  be  most 
simply  represented  by  shifting  the  resulting  supporting  force  N 


FIG.  21. — Contact 
in  Rest. 


FIG.  22. — Resistance 
to  Rolling. 


FIG.  23.— Wheel 
on  Road. 


through  a  small  distance  d  toward  the  side  of  advance,  as  in  Fig. 
22.  This  force  now  passes  the  center  of  the  roller  at  the  distance 
*/,  and  exerts  the  moment  Nd  or  Qd,  against  the  turning.  If 
instead  of  a  roller  on  a  firm,  smooth,  clean  track  we  have  a  wheel 
on  a  road  covered  with  dust  or  mud,  sand  or  loose  stone,  this 
effect  is  very  much  greater,  as  indicated  by  Fig.  23. 

34.  A  long- accepted  approximation  for  the  value  of  d  is  that 
it  may  be  taken  as  a  constant  at  0.02  inch,  being  independent 
of  the  diameter  of  the  roller,  and  also  of  the  material,  provided 
that  the  loading  is  done  with  due  regard  to  the  strength  of  the 
latter.  The  low  resistance  obtained  with  small  diameters  in 
ball  and  roller  bearings  indicates  much  smaller  values  of  d  for 
these  arrangements.  We  shall  not  attempt  any  general  dis- 
cussion of  this  question  here;  but  for  the  problems  in  the  present 


24 


GRAPHICS  OF  MACHINE  FORCES 


course  shall  assume  the  constant  value  d=o.o2  inch.  Note 
that  if  the  machine  is  drawn  to  a  reduced  scale,  the  distance  d 
must  be  divided  accordingly. 

35.  The  typical  conditions  under  which  rolling  resistance  will 
enter  into  a  graphical  determination  are  represented  in  Figs.  24 
to  26.  In  Fig.  24  the  driving  force  P  passes  through  the  center 
of  the  roller  and  is  parallel  to  the  track;  then  at  the  track  surface 
there  is  developed  an  equal  and  opposite  holding  friction  T,  which 
is  a  resistance  to  slipping  at  the  contact;  and  the  moment  PR 
is  equal  to  Qd.  The  diagonal  OC  is  the  common  line  of  the  re- 
sultants of  Q  and  P  and  of  N  and  T,  and  serves  to  determine 
the  value  of  P  in  force  diagram  b. 


FIG.  24. — Moving 
Force  at  Center. 


FIG.  25.— Wheel 
with  Journal. 


C  T 

FIG.  26.— Case  of 
Double  Rolling. 


Fig.  25  shows  the  actual  wheel,  with  the  load  Q  impressed 
through  a  bearing  and  journal.  The  force  P  must  now  over- 
come both  rolling  resistance  and  journal  friction,  and  the  re- 
sultant line  OC  has  the  added  inclination  due  to  the  radius  of 
the  friction  circle,  measured  horizontally  toward  the  side  away 
from  d. 

In  Fig.  26  is  represented  the  case  of  double  rolling,  the  roller 
lying  between  two  flat  surfaces;  this  is  typical  of  ball  and  roller 
bearings.  The  displacement  d  is  now  measured  off  at  both 
contacts,  in  opposite  directions,  but  the  resultant  line  has  the 
same  inclination,  and  the  force  P  the  same  value  relative  to  Q, 
as  in  Fig.  24.  This  force  is  the  same,  with  double  resistance, 
because  its  moment-arm  is  twice  what  it  was  in  Fig.  24;  the 
equation  of  moments  is  now  PX2R  =  QX2d.  From  another 


TOOTHED-GEAR,  CHAIN,  AND  ROPE  RESISTANCES    25 


point  of  view,  note  that  in  one  revolution  of  the  roller  P  moves 
2xR  in  Fig.  24  and  ^xR  in  Fig.  26,  doing  twice  as  much  work 
in  the  second  case,  and  thus  overcoming  the  two  rolling  resist- 
ances, each  over  the  angular  distance  2~. 


G.  Toothed-gear,  Chain,  and  Rope  Resistances 

36.  The  fundamental  conditions   as   to   the  transmission  of 
pressure  by  gear  teeth  are  set  forth  in  Fig.  27.     With  involute 


FIG.  27. — Line  of  Tooth 
Thrust. 


FIG.  28. — Inclination  of 
Thrust  Line. 


profiles,  case  a,  the  locus  of  the  point  of  contact  is  a  straight 
line  AB,  at  an  angle  of  about  75  deg.  *  with  the  line  of  centers, 
and  this  is  also  the  line  of  tooth  thrust.  With  cycloidal  pro- 
files, case  b,  the  corresponding  locus  is  a  double  curve  ACB, 
made  up  of  arcs  of  the  two  describing  circles;  and  the  line  of 
thrust  is  continually  oscillating  from  CD  at  90  deg.  to  CE  at 
perhaps  60  deg.  with  the  line  of  centers.  For  graphical  pur- 
poses we  assume  either  that  gears  have  involute  teeth,  or  that 
the  action  of  cycloidal  teeth  may  be  well  enough  represented 
by  an  average  line  of  thrust,  constant  at  75  deg. 


*  In  some  systems  the  angle  is  not  exactly  75  deg.,  but  here  we  shall  use  this 
angle. 


26 


GRAPHICS  OF  MACHINE  FORCES 


37.  Fig.  28  illustrates  a  simple  rule  for  fixing  the  direction 
in  which  the  line  of  thrust  is  deflected  from  the  tangent  to  the 
pitch  circles.     If  from  the  pitch  point  C  we  run  out,  in  the  di- 
rection of  motion,  along  the  tangent  CD,  the  thrust-line  AB— 
or,  to  be  absolutely  exact,  the  part  CB — will  be  swung  away 
from  the  driver  and  toward  the  driven  gear.     A  few  trials  of 
possible  cases,  with  sketched-in  profiles  of  the  sides  of  teeth  in 
action,  will  show  that  this  always  holds. 

38.  The  teeth  of  a  pair  of  gears  slide  together  as  they  ap- 
proach the  pitch-point,  slide  apart  as  they  recede  from  it.     The 
friction  action  which  accompanies  this  sliding  is  illustrated  in 


DRIVEN. 

FIG.  29. — Friction  on  Gear  Teeth. 


Fig.  29,  which  is  supposed  .to  represent  general,  average  condi- 
tions. The  dot-and-dash  line  AB,  passing  through  the  pitch- 
point,  is  the  line  of  thrust  for  the  case  of  no  friction.  At  the 
contacts  A  and  B  are  made  the  regular  determinations  for  the 
deflection  due  to  sliding  friction,  resulting  in  the  pressure-lines 
AF  and  BG.  These  lines  meet  at  D;  and  the  pressures  being 
assumed  equal  and  laid  off  in  DF  and  DG,  have  a  resultant 
DH  which  is  parallel  to  AB. 

39.  The  distance  DE,  through  which  the  line  DH  is  shifted 
from  AB,  is  found  as  follows: 

Angle  DAB  or  DBA  is  the  friction-angle  a;  distance  AB  is 
essentially  the  same  as  the  circular  pitch  of  the  teeth,  which  we 


TOOTHED-GEA  R,  CHAIN,  AND  ROPE  RESISTANCES    27 

shall  call  /;    and  letting  /*=tana  be  the  coefficient  of  friction 
as  heretofore,  we  have, 


(19) 


The  general  principle  set  forth  in  paragraph  25,  that  friction 
must  hinder  motion,  decides  that  this  shift  5  will  always  be  away 
from  the  driver  and  toward  the  driven  wheel;  thus  giving,  in 
Fig.  29,  a  greater  lever-arm  to  the  resistance  (3  on  2),  and  de- 
creasing the  lever-arm  of  the  driving  force  (2  on  3). 

40.  If  a  chain  carrying  a  load  runs  over  a  sheave  or  pulley, 
as  in  Fig.  30,  there  is  developed  a  certain  amount  of  frictional 


FIG.  30. — Friction  in  Hoisting 
Chain. 


FIG.  31. — Action  of  Friction 
between  Links. 


resistance  due  to  the  turning  of  the  links  upon  each  other  at 
the  places  where  the  chain  runs  on  and  off  the  pulley.  At  the 
"  on  "  side,  the  tension  Q  is  a  load,  and  the  friction  causes  it 
to  be  moved  outward,  farther  from  the  center  of  the  pulley;  at 
the  "  off  "  side,  the  tension  P  is  the  driving  force,  hence  it  is 
shifted  inward  and  its  moment-arm  decreased;  and  to  agree 
with  these  effects,  friction  at  the  journal  causes  the  bearing- 
pressure  B  to  be  moved  away  from  Q  and  toward  P.  Fig.  31 
shows  that  the  turning  of  one  link  in  another  is  that  of  a  pin 


28  GRAPHICS  OF  MACHINE  FORCES 

within  a  very  loose  eye;  and  that  there  is  a  tendency  for  the  pin 
to  hang  or  climb  on  the  side  of  the  bearing,  thus  giving  to  the 
force-line  a  deflection  greater  than  the  radius  of  the  friction 
circle  for  the  pin.  The  amount  of  the  sidewise  motion  is  de- 
termined by  the  requirement  of  making  the  force-line  come 
tangent  to  two  friction  circles,  one  for  pin  and  one  for  eye.  This 
effect  can  be,  and  is,  most  simply  taken  into  account  by  using 
an  abnormally  large  coefficient  of  friction,  with  the  diameter 
of  the  round  bar,  or  "  pin,"  as  that  of  the  equivalent  close  turning 
joint. 

41.  The  condition  just  described  (represented  in  Fig.  30) 
belongs  to  the  case  where  the  pulley  merely  changes  the  direc- 
tion of  the  chain;  except  for  friction,  the  tensions  P  and  Q  are 
the  same.  When,  however,  the  chain  exerts  a  turning  effort 
on  the  pulley  (or  vise  versa),  there  is  an  action  analogous  to  the 
sliding  of  gear  teeth,  and  a  corresponding  friction  effect.  With 
a  round-bar  chain,  pockets  are  formed  in  the  wheel-rim,  into 
which  the  links  settle;  with  flat-link  transmission  chains,  toothed 
sprocket  wheels  are  used.  If  chain  and  wheel  fit  properly,  they 
will  run  smoothly,  and  the  friction  on  pocket-edges  or  teeth 
can  be  added  to  that  of  the  chain  joints,  merely  increasing  the 
displacement  of  the  force-lines.  If  through  wear  in  its  joints 
the  chain  stretches,  the  tooth  action  will  become  very  irregular, 
and  only  the  average  effect  can  be,  perhaps,  roughly  calculated 
or  represented. 


FIG.  32. — Rope  Resistance. 

42.  When  a  rope  runs  around  a  pulley,  as  depicted  in  Fig. 
32,  there  is  at  the  "  on  "  and  "  off  "  points  a  bending  or  straighten- 
ing which  can  take  place  only  by  the  sliding  of  the  fibres  upon 


BELT   TRANSMISSION  29 

each  other,  and  which  is  resisted  by  the  friction  of  these  fibres. 
The  effect  is  to  shift  the  "  on  "  force  Q  outward  and  the  "  off  " 
force  P  inward,  just  as  in  Fig.  30.  The  displacement,  for  hemp 
ropes,  is  given  by  the  empirical  formula, 

(20) 


d  being  the  diameter  of  the  rope  in  inches.  Note  that  this  s 
must  always  be  worked  out  with  the  full-size  value  of  d,  not 
with  d  as  scaled  directly  from  a  reduced  drawing,  and  then 
brought  down  to  scale.  The  value  of  s  will  vary  with  the  kind 
of  rope,  with  age  and  condition,  and  perhaps  with  the  tension; 
but  for  the  problems  in  this  course  the  formula  just  given  will 
be  considered  as  of  universal  application. 

H.  Belt  Transmission 

43.  The  manner  in  which  power  is  transmitted  by  a  belt  is 
represented  in  Fig.  33.  In  a  state  of  rest  the  belt  is  simply 
stretched  around  the  pulleys,  and  there  is  the  same  tension  TO 
on  both  "  sides."  When  pulley  2  drives  pulley  3,  the  tension 
in  the  tight  side  (which  approaches  2)  rises  to  Ti9  while  that  in 
the  slack  side  (running  toward  the  driven  pulley)  falls  to  T2. 
The  effective  force  transmitted  is  the  difference  of  tensions, 

P=(T1-T2) (21) 

With  moderate  loading,  or  so  long  as  P  remains  well  within  the 
capacity  of  the  belt,  the  sum  of  the  tensions  remains  nearly  con- 
stant, or, 

2)  =  2r0; (22) 


but  if  the  belt  is  overloaded  and  made  to  slip  at  a  considerable 
rate,  this  sum  increases.  The  effective  radii  for  these  forces 
should  be  measured  to  the  middle  of  the  belt  thickness:  with 
crowned  pulleys,  it  is  usually  accurate  enough  to  take  the  radius 
of  the  pulley  at  the  middle  of  its  width.  Strictly,  there  is  a 


30  GRAPHICS  OF  MACHINE  FORCES 

stiffness  action  in  a  belt,  like  that  shown  for  a  rope  in  Fig.  32; 
but  the  effect  is  relatively  insignificant,  unless  with  thick  belts 
on  small  pulleys,  and  we  shall  disregard  it. 

44.  In  Fig.  33  (and  in  all  our  present  problems)  it  is  as- 
sumed that  the  belt  is  stretched  to  a  straight  line,  or  that  there 
is  no  sag  due  to  weight.  The  two  belt  lines  are  prolonged  to 
their  intersection  at  A,  from  this  point  the  tensions  T\  and  T2 
are  laid  off,  and  the  resultant  is  found.  This  resultant  is  the 
pressure  on  the  bearings,  due  to  belt  pull;  also,  it  exerts  on  each 
pulley  the  same  turning  moment  as  does  the  effective  force  P 
at  the  circumference.  In  any  problem  in  belt  gearing  we  wish 
to  locate  and  determine  this  resultant;  to  do  so,  we  need  to  know 
the  ratio  of  Tl  to  T2. 


FIG.  33. — Action  of  Belt  Drive. 

45.  The  belt  is  kept  from  slipping  around  on  the  pulley  by 
friction — but  this  is  a  holding  friction  which  prevents,  not  merely 
opposes,  relative  movement.  Referring  to  Fig.  12  and  para- 
graph 15,  we  may  compare  belt  friction  to  the  holding  friction, 
less  than  fiN,  which  balances  a  force  that  is  not  large  enough  to 
overcome  the  total  friction  and  produce  motion.  The  condi- 
tions as  to  pressure  and  friction  between  belt  and  pulley  are 
illustrated  in  Fig.  34.  Diagram  a  reproduces  the  driven  pulley 
from  Fig.  33.  Over  the  arc  from  the  on  point  A  to  the  off  point 
B  the  tension  increases  at  a  continuous  rate  from  T2  to  T\;  and 
at  any  position  located  by  the  angle  a  its  value  may  be  repre- 
sented by  the  general  (variable)  symbol  T.  The  unit  pressure 
n  between  belt  and  pulley  will  vary  with  T,  as  will  the  friction 
fin;  and  the  total  friction  developed  will  be  equal  to  Ti  —  T2. 


BELT   TRANSMISSION 


31 


46.  Consider  now  conditions  over  an  infinitesimal  arc  Rda, 
as  represented  in  Fig.  346.  The  normal  pressure  dN  on  this 
length  (or  area)  is,  from  the  force  diagram  drawn, 

dN=[T+  (T+dT)]  sin  Jda 


.....     (23) 

the  simplifications  of  the  expression  being  exact  at  the  limiting, 
infinitesimal  condition.  The  friction  pdN  is  equal  to  dT,  whence 
we  have, 

......      (24) 


FIG.  34. — Action  of  Belt  Friction. 

Integrating  from  zero  to  a,  or  from  OA  to  OB  on  Fig.  340,  we 
have, 


•- 

J  T, 


dT  T\ 

-^T  =  1°ge^"==//a; 

1  1  2 


or, 


r. 


(25) 


Here  e  is  the  base  of  the  Naperian  logarithms,  and  a.  is  expressed 
in  radian  units. 

47.  The  relation  given  in  equation  (25)  is  put  into  graphical 
form  by  means  of  the  logarithmic  spiral  drawn  in  Fig.  35.     The 


32 


GRAPHICS  OF  MACHINE  FORCES 


radius  OA  is  taken  as  unity,  since  for  a  =  o  the  ratio  r=i;  and 
the  origin  of  the  curve  is  at  OA.  For  the  coefficient  of  belt 
friction,  //=o.28  is  used  as  a  good  average  value.  Any  radius 
vector  OB  shows  the  value  of  r  for  the  contact-angle  AOB. 
With  a  pair  of  unequal  pulleys  the  arc  of  contact  on  the  smaller 
is  what  determines  the  ratio  of  T\  to  7"2;  with  an  open  belt 
(under  the  assumption  of  no  sag)  this  arc  will  never  exceed  a 


FIG.  35. 

half-circle,  but  with  a  crossed  belt  it  will  be  greater.  In  locating 
a  radius  OB  on  Fig.  35,  it  is  well  to  measure  off  the  acute-angle 
supplement  DOB  (see  also  Fig.  340)  rather  than  the  obtuse 
angle  AOB.  Generally,  the  ratio  thus  found  will  be  used  to 
fix  the  line  of  the  resultant  belt-pull,  and  after  that  force  has 
been  determined  it  will  be  resolved  back  into  components  in 
order  to  get  the  individual  tensions.  To  locate  the  resultant- 
line,  lay  off  any  convenient  length  on  the  line  of  T%  and  r  times 


GENERAL  PROCEDURE  33 

this  length  on  the  line  of  TI,  and  complete  the  parallelogram, 
as  in  Fig.  33. 

48.  What  has  been  given  is  only  the  beginning  of  a  complete 
theory  of  belt  action,   but  it  is  enough  for  present  purposes. 
In  all  problems  involving  belt  gearing,  we  will  assume  that  the 
relation  of  load  to  tensions  corresponds  to  an  existing  friction- 
coefficient  of  the  value  0.28,  and  will  use  Fig.  35  to  find  the 
ratio  of  tensions,  after  the  manner  just  indicated.     The  value 
of  r  can  be  read  off  accurately  enough  (to  two  places  of  decimals) 
with  the  help  of  the  circular  scale-lines  on  the  diagram.     At 
1 80  deg.  the  value  of  r  is  2.41. 

I.  General  Procedure 

49.  In  solving  a  problem  which  calls  for  the  determination 
of  the  forces  in  a  machine,  proceed  as  follows: 

(a)  Begin  with  the  piece  on  which  the  known  force  acts, 
whether  driving  force  P  or  load  force  Q.     Determine  the  forces 
on  this  piece,  then  go  to  the  next,  and  so  on  till  the  last  piece 
is  reached. 

(b)  In  considering  any  piece,  note  the  other  machine-members 
which  touch  it  in  sliding  or  turning  joint  or  other  contact,  and 
remember  that  each  of  these  impresses  a  force  upon  the  piece 
under  consideration. 

(c)  Go  through  the  whole  force  construction  without  friction 
first.     On  the  diagrams  mark  clearly  the  directions  of  all  forces, 
showing  for  each  piece,  in  the  individual  triangle  or  portion  of 
the  diagram,  the  forces  which  act  upon  the  piece,  not  its  actions 
outward. 

(d)  Having  thus  found  the  general  directions  of  all  forces, 
mark  the  "  contacts  "  after  the  manner  of  Fig.  13   or  Fig.    17, 
determine  and  mark  the  relative  movements  (by  inspection  or, 
if  necessary,  by  the  method  of   Fig.   19),  and   show  the   incli- 
nations of  the  force-lines  with  friction. 

(e)  In  getting  friction-circle  diameters  by  equation  (10)  and 
in  calculating  the  displacement  of  gear-tooth  pressure  by  equa- 
tion  (19),  the  scale  of  representation  may  be  disregarded,  and 


34  GRAPHICS  OF  MACHINE  FORCES 

4 

journal  radius  or  tooth  pitch  be  used  as  measured  directly  from 
the  drawing.  But  for  rolling  resistance  and  rope  stiffness,  note 
the  remarks  in  paragraphs  34  and  42. 

(/)  Belt  friction  is  not  classed  with  the  harmful  or  wasteful 
resistances,  and  in  the  no-friction  case  is  supposed  to  have  its 
full  effect.  The  same  resultant-line  is  used  both  without  and 
with  friction  in  the  machine. 

(g)  Work  should  be  accurate  and  neat,  the  lines  being  drawn 
with  a  hard  pencil,  sharpened  to  a  fine  chisel-edge.  See  that  the 
T-square  is  straight.  Measure  as  accurately  as  possible.  A 
triangular  scale  with  graduations  from  10  to  60  per  inch  will  be 
found  most  convenient. 

(/z)  Use  judgment  in  placing  force  diagrams.  Sometimes  it 
is  better  to  draw  them  right  on  the  figure  of  the  machine,  some- 
times separate  diagrams  are  preferable.  Use  different  kinds  of 
lines  for  the  forces,  as  in  Figs.  14  and  20,  namely,  dot-and-dash 
for  no  friction,  full  lines  for  forward  motion,  dotted  lines  for 
reversed  motion. 


J.  Special  Force  Constructions 

50.  In  contrast  with  the  forces  impressed  upon  a  machine 
piece  by  its  neighbors,  such  forces  as  weight  and  inertia,  de- 
pendent upon  the  mass  of  the  body  itself,  may  be  called  self- 
developed.  The  action  of  gravity  is  readily  known,  but  is  usu- 
ally of  minor  magnitude,  except  in  very  heavy  and  slow-moving 
machinery;  the  determination  of  inertia  forces,  often  of  large 
magnitude,  is  more  difficult,  and  belongs  to  the  subject  of  dy- 
namics, which  logically  comes  after  the  present  course.  We  shall 
now  briefly  consider  several  methods  which  are  especially  useful 
for  problems  involving  forces  of  this  class,  but  which  properly 
come  here  as  extensions  of  the  methods  given  in  section  B.  It 
must  be  clearly  kept  in  mind  that  so  far  as  graphical  or  mathe- 
matical relations  are  concerned,  the  kind  of  force  which  acts  is 
immaterial;  the  constructions  now  to  be  presented  are  of  uni- 
versal applicability,  but  are  most  likely  to  be  called  for  in  problems 
into  which  these  non-impressed  forces  enter. 


SPECIAL  FORCE  CONSTRUCTIONS 


35 


51.  In  Fig.  36,  AB  or  2  and  BC  or  3  are  two  links  or  members 
of  a  mechanism,  and  upon  them  act  the  known  transverse  forces 
F2  and  F%  respectively.  We  wish  to  determine  the  other  forces 
on  these  links,  for  equilibrium,  namely,  the  pin-pressures  or 
impressed  forces  at  A,  B,  and  C. 

For  link  2  take  an  origin  of  moments  at  A;  then  the  moment 
MZ  of  the  force  F^  about  A  must  be  balanced  by  the  moment 
of  the  unknown  force  (3  on  2).  Similarly,  with  an  origin  at 
C  for  link  3,  the  moment  MS  of  F%  must  be  balanced  by  that 
of  (2  on  3).  Since  (3  on  2)  and  (2  on  3)  are  equal  forces,  it  is 
required  that  their  line  of  action  pass  the  centers  A  and  C  at 


I 

FIG.  36. — Linkage  with  Transverse  Forces. 


distances  which   are  respectively  proportional   to   the  moments 
M2  and  MS.    To  find  this  line,  proceed  as  follows: 

By  measurement  and  computation,  get  numerical  values  for 
these  moments,  and  to  a  convenient  scale  lay  off  AG  =  M2 
and  CH  =  M3,  making  these  two  lines  parallel  to  each  other  and 
somewhere  near  perpendicular  to  AC.  Join  CA  and  HG,  and 
produce  them  to  meet  at  K;  then  KB  is  the  force-line  sought, 
since,  with  AL  and  CM  drawn  perpendicular  to  KB, 


(26) 


Having  found  this  line  of  force  at  joint  B,  we  get  determinate 
conditions  for  both  pieces,  2  and  3. 


36  GRAPHICS  OF  MACHINE  FORCES 

52.  In  using  Fig.  36  there  must  be  a  clear  reason  for  laying 
off  AG  and  CH  either  on  the  same  side  or  on  opposite  sides  of 
AC.     Since   (3  on  2)   and   (2  on  3)   are  opposing  forces,  their 
moments  will  be  opposite  if  the  arms  AL  and  CM  are  on  the  same 
side  of  AC,  but  alike  if  these  lever-arms  are  opposite:    conse- 
quently, if  the  original  forces  have  opposing  moments  as  in  Fig. 
36,  the  lines  AG  and  CH  are  to  be  drawn  in  the  same  direction; 
but  if  the  moments  M%  and  MS  tend  to  produce  rotation  in  the 
same  direction,  their  representing  lines  must  be  opposite.     The 
distant-intersection  constructions  of  Figs.  6  and  7  will  often  be 
found  useful  in  this  connection. 

53.  Referring  to  Fig.  i  and  paragraph  4,  we  see  that  in  order 
to  have  a  determinate  condition  for  a  body  under  the  action  of 
three  forces  we  must   know  one   force  completely,   the  line  of 
another,  and  at  least  a  point  (not  the  intersection)  through  which 
the  third  force  must  pass.     In  Fig.  36  we  have,  for  each  link,  one 
force  fully  known,  but  only  a  point  on  the  line  of  each  of  the  other 
two;  and  in  the  example  there  presented  two  such  indeterminates 
are  combined  to  produce  a  definite  condition.     It  is  now  in  order 
to  consider  a  little  more  fully  this  combination  of  one  known 
force  and  two  points  of  application. 

54.  In  Fig.  37,  the  link  or  rod  AB  is  acted  upon  by  the  known 
force  F,  while  the  other  two  forces  are  to  act  through,  or  be  ap- 
plied at,  the  pin-centers  A  and  B.     The  lines  of  these  two  forces 
must  meet  on  the  line  of  F,  but  the  intersection  may  be  anywhere 
on  that  line;    there  is,  consequently,  an  infinite  number  of  pos- 
sible pairs  of  such  forces.     In  the  figure,  two  points  are  chosen, 
at  D  and  at  E,  and  for  each  the  force  F  is  measured  off  and 
resolved,   to   get  its  components  or   equilibrants   at  A   and   B. 
These  forces,  PA  and  PB,  are  then  carried  out  to  A  and  B,  and 
laid  off  from   these  respective  points,   on   the  lines  of  action. 
Further,  the  forces  are  at  A  and  B  resolved  into  two  components, 
one  parallel  to  F,  the  other  along  the  line  of  centers  AB ;  and  the 
very   important  relation   appears   that   the  parallel  components 
AM  and  BN  are  constant,  while  those  along  the  line  AB  balance 
each  other. 


SPECIAL  FORCE  CONSTRUCTIONS 


37 


55.  By  drawing  GK  parallel  to  AB  in  the  force  triangle  DGH, 
and  noting  the  similarity  of  triangles  DGK,  DAC,  and  of  HGK, 
DBC,  we  have, 


DK:DC:  :GK:AC, 


and 


HKiDC:  :GK:BC; 


from  which  we  can  get; 


DK:HK:  :BC:AC. 


(27) 


Since  the  lengths  AC  and  BC  are  fixed,  it  follows  that  the  con- 
stant force  F  is  divided  at  K  in  a  fixed  ratio;  therefore  the  parts, 


FIG.  37. — Two-joint  Link  with  Transverse  Force. 

DK  equal  to  AM  and  HK  equal  to  BN,  are  constant.  Further, 
since  AM  and  BN  are  in  the  inverse  ratio  of  their  distances  AC 
and  BC,  they  are,  as  is  apparent  on  the  mere  inspection  of  the 
diagram,  nothing  but  the  parallel  components  of  .F  at  A  and  B, 
or  the  equilibrants  of  those  components. 

56.  What  has  just  been  shown  by  geometrical  analysis  can, 
perhaps,  be  more  simply  set  forth  as  follows: 

The  forces  applied  at  A  and  B  must  balance  F  and  also 
balance  each  other,  exerting  one  effect  parallel  to  F,  the  other 
along  the  line  AB.  Each  one  must  be,  therefore,  compounded 


38  GRAPHICS  OF  MACHINE  FORCES 

of  the  equilibrium  component  of  F,  either  AM  or  BN,  and  a 
component  along  the  line  AB.  If  we  draw  a  line  through  M 
parallel  to  AB,  all  vectors  from  A  representing  possible  values 
of  the  pressure  PA  will  have  their  ends  on  this  line  MQ;  and 
NR  is  a  similar  locus  for  PB. 

Having  then  the  force  F  on  the  link  AB,  we  divide  F  into 
parallel  components  at  A  and  B;  and  lines  drawn  through  the 
ends  of  these  forces  and  parallel  to  the  center-line  express,  in 
graphical  form,  the  requirement  which  the  applied  forces  PA 
and  PB  must  meet  in  order  to  produce  equilibrium.  The  force- 
action  on  the  body  is  still  indeterminate,  but  it  is  now  in  shape 
to  respond  at  once  to  the  imposition  of  a  determining  require- 
ment from  without. 


FIG.  38. — Method  of  Fig.  37  Applied  to  Problem  in  Fig.  36. 

57.  Fig.  38  shows  how  the  locus  device  just  described  can 
be  used  for  finding  the  forces  at  the  joint  A  in  Fig.  36.  Con- 
sider force  (3  on  2) :  it  is  a  partial  equilibrant  of  F2,  and  if  we 
measure  off  the  component  BM  against  the  direction  of  F2, 
force  (3  on  2)  must  be  a  vector  from  center  B  to  some  point 
(not  yet  known)  on  the  line  ME.  Again,  force  (3  on  2)  is  a 
direct  component  of  F%,  wherefore  BN  is  laid  off  in  the  direction 
of  FZ,  and  (3  on  2)  must  be  a  vector  from  B  to  some  point  on 
the  line  NG.  The  intersection  at  D  gives  a  vector  which  satis- 
fies both  requirements,  and  thus  fixes  the  force  (3  on  2).  The 
dotted  Icci  HK  and  HL  show  the  same  determination  for  the 


SPECIAL  FORCE  CONSTRUCTIONS  39 

force  (2  on  3).  This  scheme  involves  less  graphical  work  than 
that  of  Fig.  36,  but  confusion  as  to  the  sides  of  the  center-lines 
on  which  to  put  the  loci  must  be  avoided.  It  is  best  to  use  this 
construction,  and  to  check  it  by  the  relation,  based  on  para- 
graph 52,  that  if  Fz  and  FS  have  opposite  moments  about  A 
and  B,  the  line  of  (3  on  2)  will  be  outside  the  angle  ABC;  but  if 
these  moments  are  alike,  the  force-line  will  lie  within  the  angle 
ABC. 

58.  A  final  example,  in  Fig.  39,  will  sufficiently  illustrate 
the  use  of  the  method  developed  in  Fig.  37.  In  the  engine 
mechanism  there  shown,  the  pressure  of  the  wrist-pin  2  on  the 


G 

"FiG.  39. — Forces  in  Engine  Mechanism. 

connecting-rod  3  will  be  the  resultant  of  the  known  driving 
force  P  and  the  unknown  guide-bar  pressure  G.  The  weight- 
and-inertia  force  F  on  the  rod  is  known.  The  locus  EB  for 
the  rod  expresses  the  internal  requirement  for  equilibrium. 
Laying  off  P  as  WA  and  drawing  AB  parallel  to  G,  the  external 
requirement  is  expressed  by  saying  that  (2  on  3)  must  be  a  vector 
from  W  to  some  point  on  the  line  AB.  The  intersection  at  B 
gives  the  force  that  satisfies  both  requirements. 

59.  The  scheme  of  Fig.  37  would  apply  equally  well  if  the 
force  F  were  impressed  by  another  machine  piece;  and  a  three- 
joint  piece  with  a  force  such  as  weight  or  inertia  can  be  reduced 
to  the  case  of  Fig.  37  by  combining  the  latter  force  with  one  of 
the  impressed  forces,  which  at  the  leas,  must  be  known  if  con- 
ditions are  to  be  determinate. 


40  GRAPHICS  OF  MACHINE  FORCES 

60.  The  methods  of  Figs.  36  and  37  are  really  effective  only 
for  the  case  of  no  friction.     When  force-lines  must  be  tangent 
to  friction  circles  instead  of    passing  through    bearing-centers, 
exact  geometrical  relations  become  too  complex  to  be  useful. 
A   serviceable   scheme,    and   one   quite   accurate   enough,   is   as 
follows : 

Without  friction,  find  the  bearing-pressures  in  the  several 
joints  of  the  mechanism.  Using  the  methods  of  kinematics  as 
necessary,  find  the  relative  velocities  at  the  rubbing  surfaces. 
Get  the  several  work-rates  of  friction,  each  as  the  product  of 
coefficient,  pressure,  and  velocity,  sum  up  for  the  whole  machine, 
and  find  the  equivalent  force  at  the  point  of  application  of  P  or 
Q,  to  be  added  to  PQ  or  subtracted  from  Q0.  This  plan  need 
only  be  suggested  here;  its  fuller  development  belongs  to  prob- 
lems which  involve  much  of  the  dynamics  of  machinery,  such 
as,  for  instance,  the  graphical  analysis  of  engine  governors. 

K.  Friction  and  Lubrication 

61.  A  full  discussion  of  this  subject  is  inappropriate  here,, 
but  a  brief  statement  of  facts  and  principles  is  not  out  of  place. 
There  have  been  two  formulations  of  "  law  "  from  experiments. 
According  to  the  older  "  theory,"  based  chiefly  upon  Morin's 
results  which  were  published  about  1835,  the  coefficient  of  fric- 
tion is,  for  given  materials  and  lubricant,  constant  over  a  wide 
range  of  conditions;    the  coefficient  and  the  resistance  which  it 
measures  were  both  said  to  be  independent  of  area  of  contact 
and  of  velocity.     There  is  the  proviso  that  the  area  must  not  be 
so  small  as  to  make  the  pressure  excessive;    and  in  the  matter 
of  velocity,  the  resistance  is  greater  when  just  starting  and  with 
very  slowr  motion  than  at  a  fair  speed.     It  is  now  known  that 
this  hypothesis  is  good  for  dry,  smooth  surfaces  under  moderate 
pressure  and  for  the  case  of  restricted  lubrication  under  heavy 
pressure;    but  even  then  the  ratio  of  friction  to  pressure  is  a 
more  variable  quantity  than  was  supposed. 

62.  The  later  laws,  formulated  about  1885,  are  derived  from 
experiments  with  complete  lubrication — by  which  is  meant,  with 


FRICTION  AND  LUBRICATION  41 

a  film  of  oil  between  the  surfaces  and  completely  separating 
them.  The  resistance  now  to  be  overcome  is  fluid  friction,  and 
the  manner  of  its  action  is  as  follows: 

In  a  given  bearing  the  resistance  increases  but  little  with  the 
load;  or,  for  a  given  load,  the  resistance  is  almost  proportional 
to  the  areas  in  contact.  This  makes  the  coefficient  of  friction 
vary  inversely  as  the  intensity  of  pressure.  In  regard  to  the 
establishment  of  this  condition,  everything  depends  upon  the 
maintenance  of  the  oil-film;  and  this,  in  turn,  depends  partly 
upon  the  strength,  or  body,  or  toughness  of  the  oil,  and  partly 
upon  the  continuity  of  supply.  As  before,  resistance  is  high  at 
starting  and  decreases  with  speed;  but  presently  it  begins  to 
increase  with  velocity,  probably  because  the  relative  rate  of 
supply  is  less,  and  the  film  is  not  so  well  maintained.  While 
the  oil  must  have  enough  body  to  carry  the  load  without  being 
squeezed  out  of  the  bearing,  it  should  not  have  excessive  sticki- 
ness or  viscosity.  Increase  in  fluidity  is  the  reason  why  bearings 
usually  run  easier  after  they  have  become  warmed  up. 

63.  In  actual  machines,   the  conditions  of  working  will  lie 
somewhere  between  the  extremes  of  friction  in  a  constant  ratio 
to   pressure   and   friction   independent   of   pressure.     The   fric- 
tional  losses  increase  with  the  forces  transmitted,  but  at  a  slower 
rate  than  these  forces.     If  the  machine  is  well  loaded,  the  fric- 
tion is  much  less  with  full  than  with  restricted  lubrication — 
that  is,  with  flooded  bearings  as  against  a  scanty  supply  of  oil. 
This  question  of  supply  explains  the  observed  fact  that,  with  the 
same  kind  of  surfaces  and  the  same  lubricant,  friction  is  greater 
on  a  flat  slide  than  in  a  bearing,  as  remarked  in  paragraph  20. 
In  the  former  case,  the  oil  is  continually  being  squeezed  out  and 
pushed  away;   in  the  latter  it  is  carried  around  the  journal  and 
fed  in  again  on  the  side  of  approach. 

64.  With  a  moderate  supply  of  oil,  the  coefficient  of  friction 
is  likely  to  range  from  0.06  to  o.io  in  bearings,  and  two  or  three 
points  higher  in  slides,  or  from  0.08  to  0.13.'    With  full  flooding 
and  with  a  good  accommodation  of  lubricant  to  pressure,  it  may 
be  as  low  as  o.oi  to  0.02  in  well-made  bearings.     It  is  to  be 


42  GRAPHICS  OF  MACHINE  FORCES 

noted  that  geometrical  perfection  of  shape,  with  consequent 
uniformity  in  the  thickness  of  the  oil-film,  is  highly  important 
to  easy  running.  In  the  practice  problems  of  this  course  there 
is  a  natural  tendency  to  use  high  values  of  the  coefficient  of 
friction. 


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' '    Paper 50 


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JANNETTAZ,  EDWARD.     A  Guide  to  the  Determination  of  Rocks: 

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'KRAUCH,    C.,    Dr.     Testing    of    Chemical    Reagents    for    Purity. 

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MASSIE,  W.  W.,  and  UNDERBILL,   C.   R.     Wireless  Telegraphy 

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MOSES,  ALFRED  J.,  and  PARSONS,  C.  L.     Elements  9f  Mineralogy, 

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PALAZ,  A.,  Sc.D.  A  Treatise  on  Industrial  Photometry,  with  special 
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Illustrated $4.00 

PARSHALL,  H.  F.,  and  HOBART,  H.  M.     Armature  Windings  of 

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PERRY,   JOHN.     Applied  Mechanics.     A   Treatise   for  the   Use   of 

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PLATTNER.     Manual  of  Qualitative  and  Quantitative  Analysis  with 

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POPE,  F.  L.  Modern  Practice  of  the  Electric  Telegraph.  A  Tech- 
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PRELINI,  CHARLES.  Tunneling.  A  Practical  Treatise  containing 
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RANKINE,  W.  J.  MACQUORN,  C.E.,  LL.D.,  F.R.S.     Machinery  and 

Mill-work.  Comprising  the  Geometry,  Motions,  Work,  Strength,  Con- 
struction, and  Objects  of  Machines,  etc.  Illustrated  with  nearly  300 
woodcuts.  Seventh  edition.  Thoroughly  revised  by  W.  J.  Millar.  8vo, 
cloth $5.00 

—  The  Steam-Engine  and  Other  Prune  Movers.    With  diagram  of 
the    Mechanical    Properties    of    Steam.     With    folding    plates,    numerous 
tables  and  illustrations.     Fifteenth  edition.     Thoroughly  revised  by  W.  J. 
Millar.     8vo,  cloth $5.00 

-Useful  Rules  and  Tables  for  Engineers  and  Others.  With 
appendix,  tables,  tests,  and  formulae  for  the  use  of  Electrical  Engineers. 
Comprising  Submarine  Electrical  Engineering,  Electric  Lighting,  and 
Transmission  of  Power.  By  Andrew  Jamieson,  C.E.,  F.R.S.E.  Seventh 
edition,  thoroughly  revised  by  W.  J.  Millar.  8vo,  cloth $4.00 

A  Mechanical  Text-Book.      By  Prof.  Macquorn  Rankine  and  E.  E. 

Bamber,  C.E.  With  numerous  illustrations.  Fifth  edition.  8vo, 
cloth $3.50 

RANKINE,  W.  J.  MACQUORN,  C.E.,  LL.D.,  F.R,S.  Applied  Me- 
chanics. Comprising  the  Principles  of  Statics  and  Cinematics,  and  Theory 
of  Structures,  Mechanics,  and  Machines.  With  numerous  diagrams. 
Eighteenth  edition.  Thoroughly  revised  by  W.  J.  Millar.  8vo,  cloth  $5.00 

—  Civil  Engineering.      Comprising   Engineering,    Surveys,    Earthwork, 
Foundations,  Masonry,  Carpentry,  Metal-Work,  Roads,  Railways,  Canals, 
Rivers,    Water-Works,    Harbors,    etc.     With   numerous   tables   and   illus- 
trations.   Twenty-third  edition.    Thoroughly  revised  by  W.  J.  Millar. 
8vo,  cloth $6.50 

RATEAU,    A.     Experimental    Researches    on    the    Flow    of    Steam 

Through  Nozzles  and  Orifices,  to  which  is  added  a  note  on  The  Flow  of 
Hot  Water.  Authorized  translation  by  H.  Boyd  Brydon.  12mo,  cloth. 
Illustrated net,  $1.50 

RAUTENSTRAUCH,    W.,    Prof.,  and    WILLIAMS,    J.    T.     Machine 

Drafting  and: Empirical  Design.  A  Textbook  for  Students  in  Engineering 
Schools  and  Others  Who  are  Beginning  the  Study  of  Drawing  as  Applied 
to  Machine  Design.  Part  I.  Machine  Drafting.  Illustrated,  70  pp., 

8vo,  cloth net,  $1.25 

Complete  in  Two  Parts.     Part  II  in  preparation. 


STANDARD    TEXT  BOOKS.  11 

RAYMOND,  E.  B.  Alternating  Current  Engineering  Practically 
Treated.  Third  edition,  revised  and  enlarged,  with  an  additional 
chapter  on  "The  Rotary  Converter."  12mo,  cloth.  Illustrated.  232  pages, 

net,  $2.50 

REINHARDT,  CHAS.  W.     Lettering  for  Draughtsmen,  Engineers  and 

Students.  A  Practical  System  of  Free-hand  Lettering  for  Working  Draw- 
ings. New  and  revised  edition.  Thirty-first  thousand.  Oblong  boards. 

$1.00 

RICE,  J.  M.,  Prof.,  and  JOHNSON,  W.  W.,  Prof.  On  a  New  Method 
of  Obtaining  the  Differential  of  Functions,  with  especial  reference  to  the 
Newtonian  Conception  of  Rates  of  Velocities.  12mo,  paper $0.50 

RIPPER,  WILLIAM.     A  Course  of  Instruction  in  Machine  Drawing 

and  Design  for  Technical  Schools  and  Engineer  Students.  With  52  plates 
and  numerous  explanatory  engravings.  Second  edition.  4to,  cloth. $6. 00 

ROBINSON,  J.  B.  Architectural  Composition.  An  attempt  to  order 
and  phrase  ideas  which  hitherto  had  been  only  felt  by  the  instinctive  taste 
of  designers.  233  pp.,  173  illustrations.  8vo,  cloth net,  $2.50 

ROGERS,   ALLEN.     A   Laboratory   Guide   of   Industrial   Chemistry. 

Illustrated.     170  pp.     8vo,  cloth net,  $1.50 

SCHMALL,  C.  N.  First  Course  in  Analytic  Geometry,  Plane  and 
Solid,  with  Numerous  Examples.  Containing  figures  and  diagrams.  12mo, 
half  leather,  illustrated net,  $1.75- 

-  and  SHACK,  S.  M.  Elements  of  Plane  Geometry.  An  Elemen- 
tary Treatise.  With  many  examples  and  diagrams.  12mo,  half  leather, 
illustrated net,  $1 .25 

SEATON,  A.  E.,  and  ROUNTHWAITE,  H.  M.     A  Pocket-book  of 

Marine  Engineering  Rules  and  Tables.  For  the  Use  of  Marine  Engineers 
and  Naval  Architects,  Designers,  Draughtsmen,  Superintendents  and  all 
engaged  in  the  design  and  construction  of  Marine  Machinery,  Naval  and 
Mercantile.  Seventh  edition,  revised  and  enlarged.  Pocket  size. 
Leather,  with  diagrams ., $3.00 

SEIDELL,  A.     (Bureau  of  Chemistry,  Wash.,  D.  C.).     Solubilities  of 

Inorganic  and  Organic  Substances.  A  handbook  of  the  most  reliable 
Quantitative  Solubility  Determinations.  8vo,  cloth,  367  pp net,  $3.00 

SEVER,  Prof.  G.  F.     Electrical  Engineering  Experiments  and  Tests 

on  Direct-Current  Machinery.  With  diagrams  and  figures.  Second 
edition,  thoroughly  revised  and  enlarged.  8vo,  pamphlet.  Illus- 
trated   net,  $1.00 

and  TOVVNSEND,  F.  Laboratory  and  Factory  Tests  in  Elec- 
trical Engineering.  Second  Edition,  thoroughly  revised  and  enlarged. 

8vo,  cloth.     Illustrated.     236  pages net,  $2.50 

SHELDON,  S.,  Prof.,  anl  MASON,  HOBART,  B.S.  Dynamo  Elec- 
tric Machinery;  its  Construction,  Design,  and  Operation.  Direct-Current 
Machines.  Seventh  edition,  revised.  12mo,  cloth.  Illustrated. 

net,  $2.50 


12  STANDARD    TEXT    BOOKS. 

SHELDON,  S.,  MASON,  H.,  and  HAUSMANN,  E.    Alternating  Current 

Machines.  Being  the  second  volume  of  the  authors'  "Dynamo  Electric 
Machinery;  its  Construction,  Design,  and  Operation."  With  many  dia- 
grams and  figures.  (Binding  uniform  with  volume  I.)  Seventh  edition, 
completely  rewritten.  12mo,  cloth.  Illustrated net,  $2.50 

SHIELDS,  J.  E.  Notes  on  Engineering  Construction.  Embracing 
Discussions  of  the  Principles  involved,  and  Descriptions  of  the  Material 
employed  in  Tunneling,  Bridging,  Canal  and  Road  Building,  etc.  12mo, 
cloth $1.50 

SHUNK,  W.  F.     The  Field  Engineer.     A  Handy  Book  of  Practice 

the  Survey,  Location  and  Track-work  of  Railroads,  containing  a  large 
collection  of  Rules  and  Tables,  original  and  selected,  applicable  to  both  the 
standard  and  Narrow  Gauge,  and  prepared  with  special  reference  to  the 
wants  of  the  young  engineer.  Nineteenth  edition,  revised  and  enlarged. 
With  addenda.  12mo,  morocco,  tucks $2.50 

SMITH,   F.   E.     Handbook   of   General   Instruction   for   Mechanics. 

Rules  and  formulae  for  practical  men.     12mo,  cloth,  illustrated.     324  pp. 

net,  $1.50 

SOTHERN,  J.  W.  The  Marine  Steam  Turbine.  A  practical  descrip- 
tion of  the  Parsons  Marine  Turbine  as  now  constructed,  fitted  and  run, 
intended  for  the  use  of  students,  marine  engineers,  superintendent  engineers 
draughtsmen,  works  managers,  foremen,  engineers  and  others.  Third 
edition,  rewritten  up  to  date  and  greatly  enlarged.  180  illustrations 
and  folding  plates,  352  pp.  8vo,  cloth net,  $5.00 

STAHL,  A.  W.,  and  WOODS,  A.  T.  Elementary  Mechanism.  A  Text- 
Book  for  Students  of  Mechanical  Engineering.  Sixteenth  edition,  en- 
larged. 12mo,  cloth $2.00 

STALEY,  CADY,  and  PIERSON,  GEO.  S.     The  Separate  System  of 

Sewerage;  its  Theory  and  Construction.  With  maps,  plates,  and  illus- 
trations. Third  edition,  revised  and  enlarged,  with  a  chapter  on 
"  Sewage  Disposal."  8vo,  cloth $3.00 

STODOLA,  Dr.  A.  The  Steam-Turbine,  With  an  appendix  on  Gas 
Turbines  and  the  future  of  Heat  Engines.  Authorized  Translation  from 
the  Second  Enlarged  and  Revised  German  edition  by  Dr.  Louis  C.  Loewen- 
stein.  8vo,  cloth.  Illustrated.  434  pages net,  $4.50 

SUDBOROUGH,  J.  J.,  and  JAMES,  T.  C.  Practical  Organic  Chem- 
istry. 92  illustrations.  394  pp.,  12mo,  cloth net,  $2.00 

SWOOPE,  C.  WALTON.  Practical  Lessons  in  Electricity.  Princi- 
ples, Experiments,  and  Arithmetical  Problems.  An  Elementary  Text- 
Book.  With  numerous  tables,  formulae,  and  two  large  instruction  plates. 
Tenth  edition.  12mo,  cloth.  Illustrated net,  $2.00 

T1THERLEY,  A.  W.,  Prof.    Laboratory  Course  of  Organic  Chemistry, 

including  Qualitative  Organic  Analysis.  Writh  figures.  8vo,  cloth.  Illus- 
trated  net,  $2.00 

THURSO,  JOHN  W.  Modern  Turbine  Practice  and  Water-Power 
Plants.  Second  edition,  revised.  8vo,  244  pages.  Illustrated. 

net,  $4.00 


STANDARD    TEXT    BOOKS.  13 

TOWNSEND,  F.  Short  Course  in  Alternating  Current  Testing.  8vo, 
pamphlet.  32  pages net,  $0.75 

URQUHART,  J.  W.  Dynamo  Construction.  A  practical  handbook 
for  the  use  of  Engineer-Constructors  and  Electricians  in  charge,  embracing 
Framework  Building,  Field  Magnet  and  Armature  Winding  and  Group- 
ing, Compounding,  etc.,  with  examples  of  leading  English,  American, 
and  Continental  Dynamos  and  Motors.  With  numerous  illustrations. 
12mo,  cloth $3.00 

VAN  NOSTRAND'S  Chemical  Annual,  based  on  Biederman's  "  Chiem- 

ker  Kalender."  Edited  by  Prof.  J.  C.  Olsen,  with  the  co-operation  of 
Eminent  Chemists.  Revised  and  enlarged.  Second  issue  1909.  12mo, 
cloth net,  $2.50 

VEGA,  Von  (Baron).  Logarithmic  Tables  of  Numbers  and  Trig- 
onometrical Functions.  Translated  from  the  40th,  or  Dr.  Bremiker's 
thoroughly  revised  and  enlarged  edition,  by  W.  L.  F.  Fischer,  M.A.,  F.R.S. 
Eighty-first  edition.  8vo,  half  morocco . .  $2.50 

WEISBACH,  JULIUS.  A  Manual  of  Theoretical  Mechanics.  Ninth 
American  edition.  Translated  from  the  fourth  augmented  and  im- 
proved German  edition,  with  an  Introduction  to  the  Calculus  by  Eckley  B. 
Coxe,  A.M.,  Mining  Engineer.  1100  pages,  and  902  woodcut  illustrations. 

8vo,  cloth $6.00 

Sheep $7.50 

-  and  HERRMANN,  G.  Mechanics  of  Air  Machinery.  Author- 
ized translation  with  an  appendix  on  American  practice  by  Prof.  A. 
Trowbridge.  8vo,  cloth,  206  pages.  Illustrated net,  $3.75 

WESTON,    EDMUND    B.     Tables    Showing   Loss    of   Head   Due    to 

Friction  of  Water  in  Pipes.     Fourth  edition.     12mo,  full  leather.  .  .$1.50 

WILLSON,  F.  N.  Theoretical  and  Practical  Graphics.  An  Educational 
Course  on  the  Theory  and  Practical  Applications  of  Descriptive  Geometry 
and  Mechanical  Drawing.  Prepared  for  students  in  General  Science, 
Engraving,  or  Architecture.  Third  edition,  revised.  4to,  cloth, 
illustrated net,  $4.00 

—  Descriptive   Geometry,    Pure   and  Applied,  with  a  chapter  on 

Higher  Plane  Curves,  and  the  Helix.     4to,  cloth,  illustrated net,  $3.00 

WILSON,  GEO.  Inorganic  Chemistry,  with  New  Notation.  Revised 
and  enlarged  by  H.  G.  Madan.  New  edition.  12mo,  cloth $2.00 

WINCHELL,  N.  H.,  and  A.   N.     Elements  of  Optical  Mineralogy. 

An  introduction  to  microscopic  petrography,  with  descriptions  of  all 
minerals  whose  optical  elements  are  known  and  tables  arranged  for  their 
determination  microscopically.  354  illustrations.  525  pages.  8vo, 
cloth net,  $3.50 

WRIGHT,  T.  W.,  Prof.  Elements  of  Mechanics,  including  Kinematics, 
Kinetics,  and  Statics.  Seventh  edition,  revised.  8vo,  cloth $2.50 

-and  HAYFORD,  J.  F.     Adjustment  of  Observations  by  the 

Method  of  Least  Squares,  with  applications  to  Geodetic  Work.  Second 
edition,  rewritten.  8vo,  cloth,  illustrated net,  $3.00 

ZEUNER,  A.,  Dr.  Technical  Thermodynamics.  Translated  from  the 
Fifth,  completely  revised  German  edition  of  Dr.  Zeuner's  original  treatise 
on  Thermodynamics,  by  Prof.  J.  F.  Klein,  Lehigh  University.  8vo.  cloth, 
two  volumes,  illustrated,  900  pages net,  $8.00 


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